Godel's Second Incompleteness Theorem for Definable Theories

Abstract

It is proved that if $T$ is a ${\Sigma }_{n+1}$ Definable theory which is ${\Sigma }_n$-sound and extends $PA$, then $T$ can not prove the sentence ${\Sigma }_n-sound(T)$ that expresses the ${\Sigma }_n$-soundness of $T$. Optimality of this result is showed by constructing a ${\Sigma }_{n+1}$-definable and ${\Sigma }_{n-1}$-sound theory extending $PA$ such that ${\Sigma }_n-sound(T)$ is $T$-provable. It is also proved that no R.E. arithmetical theory, evevn very weak theories which are not ${\Sigma }_1$-complete, can prove ${\Sigma }_1$-soundness of itself.